例 4.1.3 (一般离散随机变量的数学期望)

设 $X$ 的概率分布为

$$X\sim \left(\begin{array}{cccc}-1& 0& 1& 3\\ \frac{1}{8}& \frac{1}{4}& \frac{3}{8}& \frac{1}{4}\end{array}\right).$$

试求 $E\left[X\right]，E\left[-X+2\right]$ 和 $E\left[X^2\right]$.

解

$$\begin{array}{rl}E[X]& =(-1)\times \frac{1}{8}+0\times \frac{1}{4}+1\times \frac{3}{8}+3\times \frac{1}{4}=1,\\ E[-X+2]& =(1+2)\times \frac{1}{8}+(0+2)\times \frac{1}{4}+(-1+2)\times \frac{3}{8}+(-3+2)\times \frac{1}{4}=1,\\ E[X^2]& =(-1{)}^2\times \frac{1}{8}+0^2\times \frac{1}{4}+1^2\times \frac{3}{8}+3^2\times \frac{1}{4}=\frac{11}{4}.\end{array}$$